Krylov methods for low-rank commuting generalized Sylvester equations

We consider generalizations of the Sylvester matrix equation, consisting of the sum of a Sylvester operator and a linear operator Π with a particular structure. More precisely, the commutators of the matrix coefficients of the operator Π and the Sylvester operator coefficients are assumed to be matrices with low rank. We show (under certain additional conditions) low‐rank approximability of this problem, that is, the solution to this matrix equation can be approximated with a low‐rank matrix. Projection methods have successfully been used to solve other matrix equations with low‐rank approximability. We propose a new projection method for this class of matrix equations. The choice of the subspace is a crucial ingredient for any projection method for matrix equations. Our method is based on an adaption and extension of the extended Krylov subspace method for Sylvester equations. A constructive choice of the starting vector/block is derived from the low‐rank commutators. We illustrate the effectiveness of our method by solving large‐scale matrix equations arising from applications in control theory and the discretization of PDEs. The advantages of our approach in comparison to other methods are also illustrated.